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I can find NOTHING limiting the card to a number of cards per household ... I find two paragraphs in the Terms than may affect, however ...
The annual fee is £125. With this offer you will receive the first year of Cardmembership with £0 annual fee. From year two of your membership, a £125 annual fee will automatically be charged to your account on or after the anniversary of your account opening date. The first year fee waiver is only available to new American Express Charge Cardmembers. A customer is not considered a new Charge Cardmember if they hold or have held in the last 6 months another American Express Charge Card.
Note where I've underlined
The 20,000 Membership Rewards bonus points will be awarded onto your account once you have been approved and you have spent and charged £2,000 within the first 3 months of Cardmembership. All introductory offers are subject to change, can be withdrawn at any time and are not available if you currently hold or have held any other Membership Rewards enrolled American Express Cards in the past 6 months.
Again, note the underlining
I have read the entire terms of service, as well as what they call their summary box, and can find nothing limiting the offer to one per household
Hope this helps
Let me know if you have questions,